On Periodic Expansions of Pisot Numbers and Salem Numbers
Klaus H. Schmidt · Bulletin of the London Mathematical Society · 1980
Let β > 1 be a real number, and let Tβ be the associated β-transformation of the unit interval [0,1) given by Tβα = βα (mod 1). We write Q for the set of rational numbers, Q (β) for the smallest sub-field of the reals containing β, and Per (β) for the set of (eventually) periodic points for Tβ, i.e. for the set of points whose orbits under Tβ, are finite. In this note we prove the following results: (1) If Q ∩ [0,1) ⊂ Per (β), then β is either a Pisot- or a Salem-number. (2) If β is a Pisot-number, then Per (β) = Q(β) ∩ [0,1). The last section contains explicit formulae for the cardinalities of the sets {Tkβα: k ⩾ 0}, α ε Q ⊂ [0, 1), if β satisfies an equation β2 = nβ + 1 with n ⩾ 1.